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A comparison study of medium-modified QCD shower evolution scenarios Thorsten Renk∗ Department of Physics, P.O. Box 35 FI-40014 University of Jyväskylä, Finland and Helsinki Institute of Physics, P.O. Box 64 FI-00014, University of Helsinki, Finland ThecomputationofhardprocessesinhadroniccollisionsisamajorsuccessofperturbativeQuan- tumChromodynamics(pQCD).Insuchprocesses, pQCDnotonlypredictsthehardreaction itself, but also the subsequent evolution in terms of parton branching and radiation, leading to a parton shower and ultimately to an observable jet of hadrons. If the hard process occurs in a heavy-ion collision, a large part of this evolution takes place in the soft medium created along with the hard reaction. Anobservationofjetsinheavy-ioncollisionthusallowsastudyofmedium-modifiedQCD shower evolution. In vacuum, Monte-Carlo (MC) simulations are well established tools to describe suchshowers. Forjetstudiesinheavy-ioncollisions,MCmodelsforin-mediumshowersarecurrently 9 beingdeveloped. However,theshower-mediuminteractiondependsonthenatureofthemicroscopic 0 degreesoffreedomofthemediumcreatedinaheavy-ioncollisionwhichistheveryobjectonewould 0 like to investigate. This paper presents a study in comparison between three different possible im- 2 plementations for the shower-medium interaction, two of them based on medium-induced pQCD n radiation, one of them a medium-induced drag force, and shows for which observables differences a between the three scenarios become visible. We find that while single hadron observables such as J RAA are incapable of differentiating between the scenarios, jet observables such as the longiudinal 8 momentum spectrum of hadrons in thejet show thepotential to do so. 2 PACSnumbers: 25.75.-q,25.75.Gz ] h p I. INTRODUCTION [23]. In medium-modified shower computations, energy - p is not simply lost but redistributed in a characteristic e way. h Jet quenching, i.e. the energy loss of hard partons cre- [ ated in the first moments of a heavy ion collision due to All current in-medium shower MC codes model the in- interactions with the surrounding soft medium has long teractionbetween partonsandthe medium ina different 2 v been regarded a promising tool to study properties of way. JEWEL (Jet Evolution With Energy Loss) [18] as- 8 the soft medium [1, 2, 3, 4, 5, 6]. The basic idea is to sumeseitherelasticcollisionswiththermalquasiparticles 1 studythe changesinducedbythemediumtoahardpro- or, to implement radiative energy loss, an enhancement 8 cess which is well-known from p-p collisions. A number of the singularpart of the partonbranchingkernels. Ya- 2 of observables is available for this purpose, among them JEM(YetanotherJetEnergy-lossModel)[19,20]makes . 1 suppression in single inclusive hard hadron spectra R the assumption that the virtuality of partons traversing AA 0 [7], the suppression of back-to-back correlations [8, 9], the medium grows according to the medium transport 9 single hadron suppression as a function of the emission coefficient qˆwhich measures the virtuality gain per unit 0 anglewith the reactionplane [10] andmostrecently also pathlength, and this medium-induced virtuality leads to : v preliminary measurements of jets have become available increased radiation. Finally, Q-PYTHIA, the code pre- i [11]. sented in [21] is a direct extension of the leading parton X energylosscomputationsdonein[4,24]andusesthedif- r Single hadron observables and back-to-back correlations ferentialradiationprobabilitiesoriginallycomputedfrom a are well described in detailed model calculations us- a single hard parton now for each parton propagating in ing the concept of energy loss [12, 13, 14], i.e. under the shower simulation. At this stage, it is hardly sur- the assumption that the process can be described by a prising that different models employ different implemen- medium-induced shift of the leading parton energy by tations of the parton-medium interaction, as the nature an amount ∆E, followed by a fragmentation process us- of this interaction crucially depends on the microscopic ing vacuum fragmentation of a parton with the reduced properties of the medium, i.e. the very thing one wishes energy. However,therearealsocalculationsfortheseob- to determine from the experiments. servables in which the evolution of the whole in-medium partonshoweris followedin ananalytic way [15, 16, 17]. A suitable strategy to determine these properties is thus Recently, also Monte Carlo (MC) codes for in-medium to study the effects of various different implementations shower evolution have become available [18, 19, 20, 21] of the parton-medium interaction for different observ- whicharebasedonMCshowersimulationsdevelopedfor ables. In this paper, we begin such a program by inves- hadronic collisions, such as PYTHIA [22] or HERWIG tigating the effects on a number of different observables resultingfromthreedifferentscenarios: Medium-induced radiation by an increase of parton virtuality dependent on the medium qˆ as used in [19, 20], an enhancement ∗Electronicaddress: [email protected].fi of the singular parts of the branching kernel leading to 2 additional radiation as used in [18, 25] and a drag force. part of PYTHIA. Momentum-dependent drag forces appear in computations modelling QCD-like N = 4 super Yang-Mills the- ories via the AdS/CFT conjecture [26], in the present A. Shower evolution in vacuum paper we use a simplified ansatz in which a parton in aconstantmediumundergoesamomentumindependent We model the evolution from some initial, highly vir- energy loss per unit pathlength. Such a drag term has tual parton to a final state parton shower as a series of not been tested in an in-medium shower evolution MC branchingprocessesa b+cwhereaiscalledtheparent code previously. → partonandbandcarereferredtoasdaughters. InQCD, Thepaperisorganizedasfollows: First,webrieflyreview the allowed branching processes are q qg, g gg and the computation ofmedium-modified hadronjet as done → → g qq. The kinematics of a branching is described in in [19, 20]. In addition, we describe the three different ter→msofthevirtualityscaleQ2andoftheenergyfraction implementations of the parton-medium interaction and z, where the energy of daughter b is given by E = zE its relation to the spacetime structure of the shower in b a andofthedaughtercbyE =(1 z)E . Itisconvenient detail. In a first comparison, we make the connection to to introduce t = lnQ2/Λ c w−here Λa is the scale previous leading parton energy loss calculations by con- QCD QCD parameterofQCD.ttakesa rolesimilarto atime inthe sidering a constant medium with fixed length. In this evolution equations, as it describes the evolution from medium, we study the energy loss of the leading parton some high initial virtuality Q (t ) to a lower virtuality and present the result in terms of energy loss probabil- 0 0 Q (t ) at which the next branching occurs. In terms ity distributions and mean energy loss as a function of m m of the two variables, the differential probability dP for the parameters characterizing the medium. In a second a a parton a to branch is [29, 30] comparison, we turn to a medium model which is closer to the experimental situation in so far as it expands and hence dilutes as a function of time. We compute various α s jet observables in this medium, such as the longitudi- dPa = Pa→bc(z)dtdz (1) 2π nal momentum distribuion inside the jet or the angular Xb,c broadening. Finally, in a last comparison we compute (as done in [19]) the suppression of the inclusive single where αs is the strong coupling and the splitting kernels hardhadronspectrumintermsofthenuclearsuppression Pa→bc(z) read factor R and compare all scenarioswith experimental AA results [7]. From this comparison, we tentatively deduce 1+z2 the relevant medium parameters. We conclude with a Pq→qg(z)=4/3 (2) discussion of the implications of the results. 1 z − (1 z(1 z))2 Pg→gg(z)=3 − − (3) z(1 z) II. MEDIUM-MODIFIED SHOWER − EVOLUTION Pg→qq(z)=NF/2(z2+(1−z)2) (4) where we do not consider electromagnetic branchings. In this section, we describe how the medium-modified N counts the number of active quark flavours for given fragmentationfunction(MMFF)isobtainedfromacom- F virtuality. putation of an in-medium shower followedby hadroniza- Atagivenvalueofthescalet,thedifferentialprobability tion. KeyingredientforthiscomputationisapQCDMC for a branching to occur is given by the integral over all shower algorithm. In this work, we employ a modifica- allowed values of z in the branching kernel as tionofthePYTHIAshoweralgorithmPYSHOW[27]. In the absence of any medium effects, our algorithm there- fore corresponds directly to the PYTHIA shower. Fur- z+(t) α thermore, the subsequent hadronization of the shower is Ia→bc(t)= dz sPa→bc(z). (5) assumedtotakeplaceoutside ofthe medium, evenifthe Zz−(t) 2π showeritselfwasmedium-modified. Itiscomputedusing the Lund string fragmentation scheme [28] which is also The kinematically allowed range of z is given by 1 M2 M2 p (M2 M2 M2)2 4M2M2 z± = 1+ b − c | a| a − b − c − b c (6) 2 Ma2 ± Ea p Ma2 ! whereM2 =Q2+m2withm thebarequarkmassorzero in the caseof a gluon. Giventhe initial parentvirtuality i i i i 3 Q2 or equivalently t , the virtuality at which the next B. Spacetime structure of the shower a a branchingoccurscanbe determined withthe help ofthe Sudakov form factor Sa(t), i.e. the probability that no While the vacuum shower evolution equations above are branching occurs between t0 and tm, where solvedinmomentumspaceonly,the interactionwiththe medium requires modelling of the shower evolution in position space as well, because the medium properties tm in a general medium change as a function of the posi- ′ ′ Sa(t)=exp dt Ia→bc(t) . (7) tion space variables. Usually, these are given in the c.m. −  Zt0 b,c frame of the collision in terms of the spacetime rapidity X   η , the radius r, the proper time τ and the angle φ, and s Thus,theprobabilitydensitythatabranchingofaoccurs knowledgeofthemediumevolutionimpliesknowledgeof at t is given by mediumpropertiessuchasthelocalmediumtemperature m T in the form T(η ,r,φ,τ). s In order to make the link from momentum space to momentum space, we assume that the average formation dP a time of a shower parton with virtuality Q is developed = Ia→bc(t) Sa(t). (8) dt   onthetimescale1/Q,i.e. theaveragelifetimeofavirtual b,c X parton with virtuality Q coming from a parent parton   b withvirtualityQ isintherestframeoftheoriginalhard These equations are solved for each branching by the a collision(the local restframe of the medium may be dif- PYSHOWalgorithm[27]iterativelytogenerateashower. ferent by a flow boost as the medium may not be static) ForeachbranchingfirstEq.(8)issolvedtodeterminethe given by scale of the next branching, then Eqs. (2)-(4) are evalu- atedto determine the type ofbranchingandthe valueof E E z, if the value of z is outside the kinematic bound given b b τ = . (10) by Eq. (6) then the event is rejected. Given t ,t and h bi Q2 − Q2 0 m b a z, energy-momentum conservation determines the rest Going beyond the ansatz of [19, 20] where we used this of the kinematics except for a radial angle by which the average formation time for all partons, in the present planespannedbythevectorsofthedaughterparentscan work we assume that the actual formation time can be be rotated. obtained from a probability distribution In order to account in a schematic way for higher order interference terms, angular ordering is enforced onto the shower, i.e. opening angles spanned between daughter τb P(τ )=exp (11) b pairs b,c from a parent a are enforced to decrease ac- (cid:20)−hτbi(cid:21) cording to the condition whichwe sample to determine the actualformationtime of the fluctuation in each branching. This establishes the temporal structure of the shower. With regard to z (1 z)b) 1 z b − > − a (9) the spatial structure, we make the simplifying assump- M2 z M2 b a a tionthatallpartonsprobethemediumalongtheeikonal trajectoryofthe showerinitiatingparton,i.e. weneglect After a branching process has been computed, the same the small difference of the velocity of massive partons to algorithmisappliedtothetwodaughterpartonstreating the speed of light and possible (equally small) changes them as new mothers. The branching is continued down of medium properties within the spread of the shower to a scaleQmin which is set to 1 GeV in the MC simula- partons transverse to its axis. tion, after which the partons are set on-shell, adjusting transverse momentum to ensure energy-momentum conservation. C. The parton-medium interaction After all possible branchings have been performed, i.e. after for all partons the condition Q Q has been Inthefollowing,weassumethatanyeffectofthemedium min ≤ reached, the resulting parton shower is connected with will affect the partonic stage of the evolution, but not a string following the Lund scheme [28] which is subse- the hadronization. This is equivalent to the idea that quently allowed to decay into hadrons. These hadrons hadronization takes place outside the medium, an as- form the observable jet, and analyzing the distribution sumptioncommonlymadealsoforleadingpartonenergy of hadrons, we may for example determine the fragmen- loss calculations. The validity of this assumption will be tation function Df→h(z), i.e. the distribution of hadron dicussed below. species h with an energy E = zE originating from a We use three different scenariosto modelthe interaction h f shower initiating parton f where E is the whole energy of partons with the medium. The first one, in the fol- f of the jet. lowing referred to as RAD, has been used previously in 4 [19, 20]. The relevant property of the medium probed The third scenario has been suggested in [18, 25]. In the is the transport coefficient qˆ(η ,r,φ,τ) which represents following, it is referred to as FMED. Here, the modifica- s the virtuality gain ∆Q2 per unit pathlength of a par- tion does not concern the parton kinematics, but rather ton traversing the medium. Note that this represents the evolution kernel, Eqs. (2–4). In this scenario, the an average transfer, i.e. a picture which would be re- singular part of the branching kernel in the medium is alized in a medium which is characterized by multiple enhanced by a factor 1+f , e.g. Eq. (2) becomes in med soft scatterings with the hard parton. However, unlike the medium in [19, 20] the virtuality transfer to a shower parton is randomizedinthe presentworksince theformationtime is distributed randomly around its average. Thus, effec- 41+z2 4 2(1+fmed) Pq→qg(z)= (1+z) tivelythepresentscenarioincludesthepossibilitytohave 3 1 z ⇒ 3 1 z − − (cid:18) − (cid:19) bothasmallformationtime andhence asmallvirtuality (14) gainandalargeformationtime correspondingto amore Theeffectofthemediumisthussummarizedinthevalue substantial increase in virtuality. off . NotethatintheFMEDscenario,noexplicitref- med In practice, we increase the virtuality of a shower par- erence to the spacetime structure of the shower is made, ton a propagating through a medium with specified in this sense, the scenario is rather different from the qˆ(η ,r,φ,τ) by other two. s Note that in the RAD scenario the shower gains energy from the medium by means of the virtuality increase, τa0+τa in the DRAG scenario the shower loses energy to the ∆Q2 = dζqˆ(ζ) (12) a medium whereas the shower energy is conserved in the Zτa0 FMED scenario. While this appears surprising at first, where the time τ is given by Eq. (11), the time τ0 is it is actually rather a matter of book-keeping. For a a a known in the simulation as the endpoint of the previous showerinthemedium,thereisnoconceptualwaytosep- branching process and the integration dζ is along the aratesoftpartonsfromtheshowerandfromthemedium. eikonal trajectory of the shower-initiating parton. If the However, the model framework outlined above does not partonisagluon,thevirtualitytransferfromthemedium treat the medium as consisting of partons, but rather as is increased by the ratio of their Casimir color factors, an effective influence on the shower. Thus, in a more 3/4 =2.25. realistic model one would define a criterion (say a mo- If ∆3 Q2 Q2, holds, i.e. the virtuality picked up from mentumscale)basedonwhichpartonsareremovedfrom a ≪ a the shower and become part of the medium. In such a themediumisacorrectiontotheinitialpartonvirtuality, we may add ∆Q2 to the virtuality of parton a before model, all three scenarios would lead to a loss of energy a from the shower to the medium through the appearance using Eq. (8) to determine the kinematics of the next of soft partons in the evolution, in addition to possible branching. If the condition is not fulfilled, the lifetime is determined by Q2 + ∆Q2 and may be significantly other mechanisms of energy transfer to the medium. a a shortened by virtuality picked up from the medium. In this case we iterate Eqs. (10),(12) to determine a self- consistent pair of ( τ ,∆Q2). This ensures that on the III. COMPARISON FOR A CONSTANT h ai a MEDIUM levelofaverages,the lifetime is treatedconsistently with the virtuality picked up from the medium. The actual lifetime is still determined by Eq. (11). In this section, we perform several computations for the In a second scenario, in the following called DRAG, we simple case of a constantmedium with fixed pathlength. assumethatthemediumexertsadragforceoneachprop- This is chiefly done in order to establish the relation of agating parton. The medium is thus characterized by a the models outlined above to older computations based drag coefficient D(η ,r,φ,τ) which describes the energy on leading parton energy loss. s loss per unit pathlength. In the simulation, the energy (and momentum) are reduced by A. Presence and absence of scaling A constant medium corresponds to a choice of a single τa0+τa value of qˆ,D or f . However, in the case of both the ∆E = dζD(ζ) (13) med a RADandtheDRAGscenario,alsothemediumlengthL Zτa0 hasto be specified, thus inprinciple the medium is char- Again,foragluontheenergylossisincreasedbythecolor acterized by two parameters. In [19] however we found factorratio2.25. Asinthe previouscase,theenergyloss anapproximatescalinglawfortheRADscenarioaccord- induced by the drag force is randomized even given the ing to whichthe modificationchiefly depends onthe vir- branching kinematics due to the randomized formation tuality picked up along the eikonal path of the shower time of a branching. initiating parton ∆Q2 = dζqˆ(ζ) or in the case of a tot R 5 E = 20 GeV E = 20 GeV d -> h+, RAD d -> h+, DRAG 100 100 vacuum vacuum ∆Q2 = 5 GeV2, L = 0.5 fm ∆E = 5 GeV, L = 0.5 fm ∆E = 5 GeV, L = 2 fm ∆Q2 = 5 GeV2, L = 2 fm ∆E = 5 GeV, L = 5 fm ∆Q2 = 5 GeV2, L = 5 fm 1 1 z) z) D( D( 0.01 0.01 0.0001 0.0001 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 z z 2 2 FIG. 1: The MMFF of a d-quark into charged hadrons for constant value of ∆Qtot =qˆL=5 GeV in the RAD scenario (left panel) and ∆Etot=DL=5 GeV in the DRAGscenario (right panel) for different pathlengths in a constant medium. constant medium simply qˆL. A similar scaling law can Thelattereffectisclearlynotrelatedtoanactualphysics alsobeestablishedfortheDRAGscenario,albeitonlyin effect but rather an artefact of the need to switch to thecaseofanexpandingmedium(seebelow). Whenever a non-perturbative description of hadronization at some suchascalinglawholds,acomparisonbetweenthediffer- point in the simulation. It is unreasonable that a parton entscenarioscanbe madebasedonthe singleparameter (or proto-hadron) would feel no effect from the medium ∆Q2 or ∆E only. just because its virtuality is small, however it is unclear tot tot It is clear that the scaling cannot work for all the pos- just how the effect should be implemented properly in sible functional forms qˆ(ζ). In two different limits this the present framework. The behaviour of the simulation canbemadeplausible: If, inthe RADscenario,∆Q2 is thusdependsontheactualchoiceofQmin,andthisneeds tot added at once initially, ∆Q2/Q2 is for reasonable values to be optimized eventually in comparison with data. A of hard process kinematics and medium properties very study of the effect of changing Qmin will be presented small. Forexample,fortypicalRHIC kinematicsthe ini- below. tialQ2 fromwhichtheevolutionstartsmaybe400GeV2 whereasthetotalvirtualityacquiredforapartontravers- The resulting MMFF for a light quark into charged ingthewholemediumisabout15GeV2 accordingtothe hadrons for constant qˆL or DL and a variation of path- results of [19]. However, such a small correction will not length is shown in Fig. 1 for both the RAD and the influencetheshowerevolutionsignificantly. Ontheother DRAG scenario. In a constant medium, the RAD sce- hand,ifthe virtualityisaddedlaterwhenthetypicalQ2 nario shows approximate scaling for pathlength between is of order of ∆Q2, a much stronger modification is ex- 0.5 and 5 fm. The DRAG scenario does not exhibit a pected. Thus, one expects the scaling law to be violated strong scaling in the region of large z, but in the region into the directionoflessmedium effectifqˆ(ζ) is strongly z 0.5 which is predominantly probed when computing ∼ peaked towards τ =0. thesinglehadronspectra,thevariationsarenottoolarge Asimilarargument,canbemadefortheDRAGscenario. for pathlengths between 0.5 and 3 fm. The drag force acts on every parton in the shower. This means that if D(ζ) is strongly peaked towards τ = 0, Notethatneithertheshortpathlengthnorthelongpath- then the drag force acts only on one parton, the shower lengthlimitisactuallyproblematicforarealisticmedium initiator, whereasif it is applied later,its effect is felt by evolutiontakenfroma hydrodynamicalmodel. The first several partons. limitisavoidedbyvirtueofthethermalizationtimeofor- On the other hand, note that the shower evolution is der O(0.6) fm for RHIC kinematics. This is a large time terminatedforeverypartonwhichreachesQ2 Q2 = compared with the timescale in which the first branch- ≤ min 1 GeV2. This implies that the typical lifetime of the ings in a shower occur, thus by the time the medium is shower for an initial parton with energy E is given by present, the shower is already well developed. The sec- τ E/Q2 ,thusashowerwithE =20GeVprobes ond limit is avoided because in an expanding medium max ∼ min the medium on average for a distance of 4 fm (Eq. (11) qˆ(ζ) or D(ζ) droprapidly as a function of time, thus the leads to fluctuations aroundthis averagethough). Thus, late time contribution to dζqˆ(ζ) or dζD(ζ) is small ifLischosenmuchbeyondτ ,qˆLorDLarenotgood in any case. Thus, the scaling works much better for a max R R parameters any more, as the shower does not effectively realistic evolution as the constant medium results would probe the whole medium. suggest. 6 B. Energy loss and quenching weights is very different for a light quark shower where the lead- ingquarkdistributiontypicallypeaksatz 0.5andany ∼ energy loss of ∆E >E/2 shifts the bulk of the distribu- We now proceed to compare the three scenarios on the tion into the unphysical region of negative energies. basis of leading parton energy loss. This is relevant to make the connection to previous calculations in the InFig.2weshowtheleadingcharmdistributionsbothin BDMPS or ASW formalism[2, 24] which are formulated vacuumandinmediumforamediumpathlengthofL=2 usingthisconcept. Forthispurpose,weselecttheshower fm. In order to make a meaningful comparison between initator to be a c-quark and extract the energy distri- the different scenarios, the average relative energy loss bution of the leading c-quark dN/dE after the shower. ∆E /E is fixed to 10% or 20% respectively. c From the comparison of the distribution dN/dEvac in h i c In order to deduce the energy loss probability distribu- vacuum and in the medium dN/dEmed, we can deduce c tion from these results, we have to solve Eq. (15). By theenergylossprobabilitydistributionP(∆E). Theidea discretizingthe integralover∆E inEq.(15) we cancast is to make an ansatz it into the form of a matrix equation dNmed dNvac ′ ′ (E)= d(∆E) (E )P(∆E)δ(E E ∆E) n dEc dEc − − N (Ei)= K (Ei,∆Ej)P (∆Ej) (16) Z i ij j (15) j=1 and solve it for P(∆E). Note that this ansatz contains X theratherdrasticassumptionthatthereisnoparametric dependenceontheinitialenergyE. IfwerequireP(∆E) where dN/dEc is provided at m discrete values of E la- tobe a probabilitydistribution, theassumptionmayim- belled Ni and P(∆E) is probed at n discrete values of ply that for some partons in the distribution dN/dEvac ∆E labelled Pj. The kernel Kij is then the calculated c theenergyloss∆E islargerthantheirenergyE inwhich dN/dEc for all pairs (Ei,∆Ej) where the energy loss case they have to be considered lost to the medium. A acts as a shift of the distribution, i.e. dN/dEcmed(E) = similar situation also occurs in the application of the dN/dEcvac(E+∆E). ASWformalismtofiniteenergykinematics. Theproblem Eq. (16) can in principle be solved for the vector P by j ofthe validity ofassumingenergyindependence however inversionofK form=n. However,ingeneralthisdoes ij only concern the comparison with the ASW results in notguaranteethattheresultisaprobabilitydistribution. which energy loss is formulated in terms of a probabil- Especially in the face of statistical errors and finite nu- ity density P(∆E). In all other results presented in this mericalaccuracythe direct matrix inversionmay permit manuscript, the full information of the shower including negative P which have no probabilistic interpretation. j finiteenergykinematicsisusedandnoassumptionabout Thus, a more promising solution which avoids the above energy independence of energy loss needs to be made. problemsistoletm>nandfindthevectorP whichmin- The choice of a c-quarkas showerinitiator has a twofold imizes N KP 2 subject to the constraints0 P 1 i motivation. First, it allows to define energy loss in the and |n| P− =1|.|This guaranteesthat the outc≤ome≤can same way as done in the ASW formalism. Note that the i=1 i beinterpretedasaprobabilitydistributionandsincethe ASW formalism assumes infinite parent parton energy P system of equations is overdetermined for m > n errors and calculates energy loss via the radiation spectrum off on individual points R do not have a critical influence i the parent. In applying the formalism to finite energy, a on the outcome any more. This is the approach we have process may occur in which a radiated gluon takes 90% chosen. of the energy of an initial quark q . This energy is then 1 consideredtobelostfromtheq . However,intheshower The results are shown for L = 2 fm in Fig. 3. Qual- 1 language, the radiated gluon would in this case become itatively, both the radiative energyloss scenarios RAD the new leading parton, and even tagging the leading andFMEDproduceenergylossprobabilitydistributions quarkoutofashowerwouldnotpreventprocesseswhere which are similar to the ASW quenching weights [24] in this gluon splits into a qq pair where the new quark q thesensethattheyareflatacrossawiderangein∆E. In 2 contrast,theDRAGscenarioproducesalocalizedpeakin mightstillbeharderastheoriginalparentq ofthegluon. 1 The choice of a c quark as shower initiator effectively theenergylossdistributionwhichremindsofthequench- ingweightsfoundforelasticenergylossscenarios[32,33]. suppressessuch processesandallowsto treatenergyloss as closely as possible to ASW [31]. Especiallyforlargerenergyloss,theRADandtheFMED scenario lead to almost identical results. The second advantage of extracting P(∆E) from a c quark is that the c-fragmentationis rather hard, i.e. the However, there is an important difference to the ASW probability distribution to find the leading c-quark af- quenching weights: While the ASW results typically ter a vacuum shower peaks close to z = 1. This effec- show a large discrete probability for no energy loss, the tivelymeansthatifoneconsidersanadditional,medium- resultsobtainedhereshownosubstantialstrengthinthe induced shift in energy, most of the energy range is still first bin (the inversion procedure outlined above cannot available for the dominant part of the distribution. This separate zero energy loss from small energy loss). 7 E = 20 GeV, <∆E>/E = 0.1 E = 20 GeV, <∆E>/E = 0.2 vacuum vacuum RAD RAD 0.12 DRAG 0.12 DRAG FMED FMED -1V] -1V] e e G0.08 G0.08 E [c E [c d d N/ N/ d d 0.04 0.04 0 0 0 5 10 15 20 0 5 10 15 20 E [GeV] E [GeV] FIG. 2: Energy distribution of theleading cquarkfor a 20 GeV cquark asshower initiator in thethreedifferentscenarios for theparton-medium interaction (see text). Left panel: 10% average energy loss, right panel: 20% average energy loss. E = 20 GeV, <∆E>/E = 0.1 E = 20 GeV, <∆E>/E = 0.2 0.8 RAD 0.8 RAD DRAG DRAG FMED FMED -1V]0.6 -1V]0.6 e e G G E) [0.4 E) [0.4 ∆ ∆ P( P( 0.2 0.2 0 0 0 5 10 15 20 0 5 10 15 20 ∆E [GeV] ∆E [GeV] FIG. 3: Energy loss probability distribution P(∆E) for the leading c-quark for a 20 GeV c-quark as shower initiator in the three different scenarios for the parton-medium interaction (see text). Left panel: 10% average energy loss, right panel: 20% average energy loss. C. Parametric dependence of mean energy loss towards some generic properties of radiative energy loss scenarios independent of the details of the implementa- tion. In order to gain more insight into the different scenarios, we investigate in Fig. 4 for a constant medium with L = 2 fm how the mean energy loss, defined as ∆E = d∆E∆EP(∆E) with P(∆E) obtained as in h i theprevioussectionbehavesasafunctionoftherelevant In particular, the RAD and the FMED scenario exhibit R medium parameters. We include a scenario in which the saturationofthemeaninducedenergylossatabout25% strong coupling constant is not allowed to run with the of the total energy as the medium effect is increased. virtuality scale in the shower (as is the default option in This saturation is even more pronounced for a constent PYSHOW) but is kept fixed at αs =0.3. αs. In striking contrast, the DRAG scenario in which There is no unique way to present and compare the re- energy is directly transferred to the medium shows an sults, asthe three relevantparametersqˆ,D andf are almostlinearrise upto meanenergylossesof50%. Note med rather different. However, as apparent from Fig. 4, it is that the extraction of the energy loss probability based possibletofindasimpleproportionalityrelationbetween on discretization and matrix inversion as outlined above qândf suchthattheriseofthe meanenergylossap- becomes increasingly problematic at ∆E /E > 0.4 due med h i pears very similar. This, in addition to the similarity of to the problemofpartonsbeing shiftedto negativeener- P(∆E)forboththeRADandtheFMEDscenariopoints gies mentioned above. 8 E = 20 GeV, L = 2 fm ton trajectory [37, 38], we find that the vast majority of 10 paths found in the 3-dimensionalhydrodynamicalmodel of Bass and Nonaka [34] leads to a qˆ(ζ) which can be RAD RAD, α = 0.3 described by the rather simple expression 8 s DRAG FMED a V] 6 qˆ(ζ)= (b+τ/(1fm/c))c. (18) e G > [ Based on this expression, we investigated three different E ∆ 4 scenarios(approximatelyrepresentingapartontravelling < into +x direction originating from x = 4 fm (A), x = 0 (B) and x = 4 fm (C), y = 0 in all cases in the trans- 2 − verse (x,y) plane at midrapidity. These trajectories are characterized by the parameters (b = 1.5,c = 3.3,τ = E 0 5.8 fm/c) (A), (b = 1.5,c = 2.2,τ = 10 fm/c) (B) and 0 5 10 15 E q [GeV2/fm] / D [GeV/fm] / 10 f (b = 1.5,c = 2.2,τE = 15 fm/c) (C) and are quite typ- med ical for partons close to the surface (A), emerging from the central region (B) or traversing the whole medium FIG. 4: Mean energy loss as a function of the medium prop- (C). As in IIIA in the present paper for a constant ertiesindifferentscenariosfortheparton-mediuminteraction medium, we found that an approximate scaling in which in a constant medium with L=2 fm. the medium effects did not depend on details of the trajectories (A), (B), or (C) but only on ∆Q2 = dζqˆ(ζ). tot IV. COMPARISON FOR A SINGLE PATH IN Thevirtueofthisscalinglawistwofold: First,itallowsto R AN EXPANDING MEDIUM presentthemediummodificationsfortherelevantclassof functions qˆ(ζ) as a function of a single parameter ∆Q2 tot We now turn to a more realistic scenario in which the only. Second, it considerablyspeeds up the computation parton propagates in a medium as created in a heavy- for a comparison with data where a weighted average ion collision. Relativistic fluid-dynamical models such over all possible paths through the medium has to be as [34] give a good description of many bulk proper- computed. ties of the medium, hence in the following we will as- In [19] we have made the rather drastic assumption that sume that hydrodynamics is a valid description of the themediumdoesnotexertanyeffectbeforethethermal- medium. Both the finite size and the finite lifetime of ization of the medium at the time τin where τin = 0.6 such a medium are felt by the parton. In particular, the fm/c in the model studied for RHIC [34]. In the fol- local density may drop a) because of a spatial variation, lowing, we adopt a more realistic approach in which we i.e. the parton reaches the medium edge and b) a tem- increase the medium effect linearly from zero at τ = 0 poral variation, i.e. the global expansion of the medium to its value reached at τin. The idea behind this is that reducestheoveralldensityasafunctionoftime. Inaddi- initially no medium can be present, as the timescale for tion, there are arguments that the hydrodynamical flow hard processes precedes any other timescale in the sys- of the medium expansionshould also have a direct influ- tem. However, even a medium which is not yet equili- ence on the medium properties as seen by the medium brated may interact with hard partons and lead to scat- due to Lorentz transformationbetween the moving local tering processes. A linear interpolation between the ini- medium rest frame and the frame of the hard collision tial time and the equilibration time seems a reasonable [37, 38]. prescription to capture part of these effects. In practice, qualitative aspects of the results of [19], in particularly the presence of the scaling, are not substantially altered A. Characterization of the medium by this modification. There is however an effect on the numerical value of extracted medium parameters. Let us now consider the other scenarios DRAG and In [19] we have established that if qˆ is linked with the FMED. Eq. (17) which links qˆ with the hydrodynami- medium properties by the relation cal properties of the medium is based on counting the potential scattering centers along the parton trajectory. qˆ(ζ)=K 2 [ǫ(ζ)]3/4(coshρ(ζ) sinhρ(ζ)cosψ) (17) ǫ3/4 for an ideal gas corresponds to the entropy density, · · − whichinturnisproportionaltothemediumdensity. The withK aparameterdeterminingtheinteractionstrength additional factor (coshρ(ζ) sinhρ(ζ)cosψ) is nothing − which is a priori unknown (in an ideal QGP, K = 1 is buttheappropriatetransformationtodeterminehowthe expected [35] but a comparisonstudy of different energy density seen by the parton is changed under a boost of loss models has shown to be inconclusive in extracting the restframe of the medium [37]. It is reasonable to as- values for K [36]), the medium energy density ǫ, the lo- sume a similar measureofpotential scatteringcentersto cal flow rapidity ρ with angle ψ between flow and par- be relevant for the other scenarios. This ansatz leads to 9 of the MMFF D(z) for three different scenarios in comparison. There is no a priori criterion at which values D(ζ)=KD·[ǫ(ζ)]3/4(coshρ(ζ)−sinhρ(ζ)cosψ) (19) of the three medium parameters ∆Q2tot, ∆Etot and fmed the three different scenarios should be compared. For for the drag coefficient D with an a priori unknown pa- thecomparisonintermsofenergylossprobabilitydistri- rameter KD specifying the overall strength of the drag butions done above we required a fixed value ∆E /E, force. but this is not a meaningful variable when onehwantis to As discussed above, the FMED scenario has no explicit compareonthe basisofthe wholepartonshowerinstead dependence on the spacetime evolution of the shower, of the leading parton kinematics only. Here, we chose but it seems reasonable the the parameter fmed should the criterion that the MMFF approximately agree in an dependonthetotaleffectofthemediummeasuredinthe interval of 0.4 < z < 0.7. This is the region of the frag- number of potential scatterers which have been encoun- mentationfunctionwhichispredominantlyprobedwhen tered. This leads to the ansatz the fragmentation function is folded with a pQCD par- tonspectrumtocomputesingleinclusivehadronproduc- tion. TheimplicationisthatacomputationwithMMFFs fmed =Kf dζ[ǫ(ζ)]3/4(coshρ(ζ) sinhρ(ζ)cosψ). agreeinginthe aboveintervalwouldyieldapproximately − Z thesameobservablehadronspectra. Thischoiceleadsto (20) theinterestingandamusingnumericalcoincidencethatif Here, as in the previous scenarios, we also introduce an the parameters are given in powers of GeV, the relation a priori unknown parameter K which determines the f ∆Q2 /GeV2 ∆E/GeV 10f holds. strength of the parton-medium interaction. tot ≈ ≈ med We show the MMFF of a 20 GeV d-quark into charged E = 20 GeV hadronsfor∆Q2 =10GeV2 (theparametersoftheother d -> h+, DRAG scenariosadjustedcorrespondingly)inFig.6,rightpanel. 100 In order to focus more on the hadron production at low vacuum ∆E = 5 GeV, (A) momenta, we introduce the variable ξ = ln(1/x) where ∆E = 5 GeV, (B) x = p/E is the fraction of the jet momentum carried ∆E = 5 GeV, (C) jet by a particular hadron and E is the total energy of jet 1 the jet. The inclusive distribution dN/dξ, the so-called z) Hump-backed plateau, is an important feature of QCD D( radiation [39, 40] and is in vacuum dominated by color coherence physics. 0.01 In Fig.6 (left panel) we showdN/dξ for the three different scenarios in comparison with the unmodified result. Itisapparentfromthefigurethatwhilethethreescenar- iosagreeinthehighzandconsequentlylowξregion,they 0.0001 0 0.2 0.4 0.6 0.8 1 exhibitsizeabledifferences inthe highξ regionwherein- z duced radiation is expected to contribute to soft hadron FIG.5: TheMMFFofa20GeVd-quarkintochargedhadrons production. Here, boththe radiativescenariosRADand for three different paths (A), (B) and (C) (see text) with FMED show the expected enhancement of the distribu- ∆Etot=5 GeV in the DRAGscenario. tion, but the DRAG scenario is strikingly different — it falls below the vacuum result. However, this is hardly The scaling within the RAD scenario of the results with surprising, as in this scenario energy is taken away from ∆Q2 has been established in [19] and in the present theevolvingshowerandishencenotavailableforhadron tot paper also for a constant medium in IIIA. The DRAG production. scenario shows no strong scaling for a constant medium, While a measurement of dN/dξ would appear to be a but as anticipated the result is more promising for an promising means to distinguish between induced radia- expanding medium. The validity of the scaling under tion and a drag force as the microscopic realization of theseconditionsisapprentfromFig.5wherewecompute energy loss, it has to be pointed out that there are two for fixed ∆E for the three different paths (A), (B) and things which urge some caution. First, the Lund scheme (C). Note that scaling of the FMED scenario is realized used to model hadronization in the present framework by definition using the ansatz Eq. (20). assumes that hadronization takes place far outside the medium. If the energy of a hadron h of mass m is E , h h thespatialscaleatwhichhadronizationoccurscanbees- B. Longitudinal momentum distribution of the timated as l E /m2. For pions, this is nota problem shower throughouthth≈e kihnemahtic range, but for kaons and pro- tons the hadronization length is considerably shortened. In Fig. 6, we show the longitudinal momentum distri- Even a 10 GeV proton has only l 2 fm, thus heavy h ≈ bution of charged hadrons inside the shower in therms hadron production in the high ξ region is not addressed 10 E = 20 GeV E = 20 GeV d -> h+- d -> h+- 10 vacuum 2 10 vacuum RAD, <∆Q2> = 10 GeV2 RAD, <∆Q2> = 10 GeV2 8 DRAG, <∆E> = 10 GeV DRAG, <∆E> = 10 GeV FMED, f = 1.0 med FMED, f = 1.0 med 1 6 ξ D(z) N/d d 4 -2 10 2 10-4 0 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 z ξ FIG. 6: Longitudinal momentum distribution of charged hadrons inside a jet originating from a 20 GeV d-quark shown as fragmentation function D(z) (left panel) and dN/dξ (right panel) for the vacuum and the three different scenarios for the parton-mediuminteraction(seetext). ThemediumparametershavebeenchosentoletthemodifiedD(z)approximatelyagree for 0.4<z<0.7 for all in-medium scenarios. adequately in the model, as one cannot safely assume E = 20 GeV d -> h+-, P > 1 GeV hadronizationtakes place outside the medium where the 8 T Lund model is applicable. Nevertheless, since pions con- vacuum stitutethebulkofchargedhadronproduction,theessen- RAD, <∆Q2> = 10 GeV2 tial features of the model are expected to be robust. 6 DRAG, <∆E> = 10 GeV Thesecondissueconcernstheeffectoftriggerbias. Ase- FMED, f = 1.0 med ries of experimental cuts has to be imposed on events in φ heavy-ioncollisionstodiscriminatehadronsbelongingto N/d4 jets from the backgroundof soft medium hadrons. How- d ever, strongly modified jets (for example those emerging fromthe medium center)are less likely to fall within the 2 cutsthanunmodifiedjets(suchasthosefromthemedium edge). Asaresultthereisatriggerbiaswhichsuppresses eventsinwhichamodificationofdN/dξ isvisible. Acal- 0 culation in the RAD scenario taking into account a re- 0 0.25 0.5 0.75 1 alistic series of experimental cuts has been performed in φ [rad] [20] and found that there should be no visible enhancement if jets are identified directly via a standard set of FIG.7: Angulardistributionofchargedhadronsabove1GeV cuts. coming from the fragmentation of a 20 GeV d-quarkfor vacuum and three different scenarios of parton-medium interaction (see text). C. Angular distribution lead to angular broadening of the jet as compared to Another possibility to identify the mechanism of the the vacuum result, whereas the DRAG scenario shows parton-medium interaction is to study the structure of no indication for broadening. the jet transverseto the jet axis. This is reflectede.g. in the angular distribution of hadrons around the jet axis. The distribution dN/dφ where φ is the angle between D. The sensitivity to Qmin hadronandjetaxisfor∆Q2 =10GeV2(theparameters tot in the other scenarios adjusted accordingly) for the vac- For a constant medium, we noted earlier that there is a uum and the three different scenarios is shown in Fig. 7 sensitivity to the choice of the minimum virtuality scale where a cut in momentum of 1 GeV has been applied Q atwhichpartonsintheshowerareevolvedfurther. min to focus on hadrons which would appear above the soft Before comparing the results of this section to data, it background of a heavy-ion collision. is reasonable to ask to what extent a choice of Q min It is apparent from the figure that the radiative energy different from its default value Q = 1 GeV has an min loss scenarios again roughly agree with each other and influence on the results.